A new flexible Weibull extension model: Different estimation methods and modeling an extreme value data

The word extreme events refer to unnatural or undesirable events. Due to the general destructive effects on society and scientific problems in various applied fields, the study of extreme events is an important subject for researchers. Many real-life phenomena exhibit clusters of extreme observations that cannot be adequately predicted and modeled by the traditional distributions. Therefore, we need new flexible probability distributions that are useful in modeling extreme-value data in various fields such as the financial sector, telecommunications, hydrology, engineering, and meteorology. In this piece of research work, a new flexible probability distribution is introduced, which is attained by joining together the flexible Weibull distribution with the weighted T-X strategy. The new model is named a new flexible Weibull extension distribution. The distributional properties of the new model are derived. Furthermore, some frequently implemented estimation approaches are considered to obtain the estimators of the new flexible Weibull extension model. Finally, we demonstrate the utility of the new flexible Weibull extension distribution by analyzing an extreme value data set.


Introduction
A number of probability methodologies (or probability distributions) have been presented and implemented to examine the main body of data and describe its behavior.These probability distributions are effectively implemented for analyzing the crises in the global financial sector, radical political events, military strikes, or natural disasters such as earthquakes, floods, and droughts [1][2][3][4][5].
However, it is also a proven and confirmed fact that no single probability distribution is apt to provide a satisfactory fit in all circumstances.Therefore, to provide a satisfactory fit to real-life scenarios, we often need new probability distributions with better and improved distributional flexibility.The needs of data modeling with satisfactory fit have motivated researchers to discover and apply new flexible probability distributions [6][7][8][9].
Considering the significance of the probability distributions in modeling the real-life events, numerous probability distributions have been suggested and implemented by improving the existing distributions.The modifications of the existing probability distributions are based on different methods.
The compounding of the probability distributions is a useful approach for developing new probability distributions.However, the new probability models obtained using the compounding approach have a difficult form of density which makes the estimation process more burdensome [10].The transformation approach, especially, the exponential transformation, is also an effective approach to obtain new models.This approach is one of the easiest methods to use for obtaining new probability distributions.However, the derivation of properties of such distributions becomes very difficult [11].Another prominent method for obtaining new distributions which provides reasonably the best-suited fit to real-life events is the finite mixture of probability distributions.This method generates new updated probability distributions with improved distributional flexibility.However, the number of parameters of the new models is also increasing.Therefore, more computational efforts and time are needed for deriving the estimators [12].
Here, we study a new probability distribution for dealing with the flood data set.The new model is introduced without incorporating new additional parameters to reduce the estimation problems and has a closed-form CDF.The new model is termed as a new flexible Weibull extension (NFWE) distribution.Despite adding no additional parameter, the NFWE distribution still performs better.

A new flexible Weibull extension model
This section introduces the basic functions of the NFWE distribution.By joining Eq. (1) with Eq. ( 2), we get the CDF of the NFWE distribution as expressed by with PDF

𝑔(𝑢
where  = ( 1 ,  2 ) ⊺ .Some behaviors of (; ) of the NFWE distribution are shown in Fig. 1, which show that as we increase  2 , the NFWE distribution tends to exhibit a longer tail.This fact reveals that due to the long right tail, the proposed NFWE distribution can be a good candidate model for dealing with the skewed data in telecommunications, finance sector, hydrology, engineering, meteorology, etc.

Distributional characteristics
Here, we provide a concise mathematical framework of the distributional characteristics of the NFWE distribution.These distributional characteristics involve the quantile function (QF), median, quartiles, skewness, kurtosis, and  ℎ moment.
The median of the NFWE model can be obtained using  = 1 2 in Eq. ( 5), given by 2 where  = log ) .
The first quartile (usually expressed by  1 ) of the NFWE distribution is where  = log ) .
The third quartile (usually expressed by  3 ) of the NFWE distribution is where  = log ) .

The 𝑟 𝑡ℎ moment
For the NFWE distribution with PDF in Eq. ( 4), the  ℎ moment is On solving, we observe where and

Parameters estimation
This section explores the frequently implemented estimation approaches to obtain the estimators ( δ1 , δ2 ) of the parameters of the NFWE distribution.These estimators include the weighted least-square (WLS) estimators, ordinary least-square (OLS) estimators, maximum likelihood (ML) estimators, maximum product spacing (MPS) estimators, Cramér-von Mises (CVM) estimators, Anderson-Darling (AD) estimators, Right-tail Anderson-Darling (RAD) estimators and percentile (PC) estimators.Recently, these methods have also been applied by [20,21] for estimating the model parameters.

The OLS and WLS estimators
Let  (1) ,  (2) , ⋯ ,  () be a set of the order statistics selected from  ( u;  1 ,  2 ) in Eq. (3).The OLS estimators (see [22]) δ1 and δ2 can be attained through minimizing with respect to  1 and  2 .We can also attain the OLS estimators δ1 and δ2 , by solving where and Furthermore, we attain the WLS estimators δ1  and δ2  , by solving where the quantities are given in Eq. ( 8) and Eq. ( 9), respectively.

The CVM estimators
The CVM estimators [25], say δ1  and δ2  , of the NFWE distribution are attained by solving where the quantities are given in Eq. ( 8) and Eq. ( 9), respectively.

The AD and RAD estimators
The AD estimators [26], say δ1  and δ2  , of the NFWE distribution are attained by solving with respected to  1 and  2 .
For the NFWE distribution, the RAD estimators, say δ1  and δ2  , are attained by solving

The PC estimators
The percentile estimation method is another most prominent method for estimating the papers [27,28].For the NFWE distribution, the PC estimators, say δ1  and δ2  , are attained by minimizing where (⋅) is the QF.
For demonstrating the performances of δ1 and δ2 , we attain different random samples:  1 ,  2 , … ,   of sizes  = 20, 50, 80, 120, 200 and 300 from the NFWE distribution by using Eq. ( 5) and simulate them  = 5000 times.The simulation is done by utilizing the  statistical software (version 4.1.1)with () function.For demonstrating the performances of δ1 and δ2 , take into account four judgment tools (also called statistical criteria).These criteria include • Absolute biases, which is computed as • Mean square errors (MSEs), which is calculated as

and
• Mean relative errors (MREs), which is obtained as Out of many simulated outcomes (thirty-six outcomes), we only report the simulation results for six combination values of  1 = {0.10,0.50, 0.95, 1.56, 2.21, 3.00} and  2 = {0.05,0.45, 0.87, 1.39, 2.15, 4.00}.For the given values of  1 and  1 , Tables 1-6 provide the simulation results of the NFWE distribution.Tables 1-6 also provide the rank of the estimators presented in the superscript indicators.The ∑  is also provided in Tables 1-6.In addition to Tables 1-6, Table 7 summarizes the partial as well as the overall ranks of  1 and  1 .Corresponding to the given facts in Tables 1-7, we can easily pick the following conclusion: Table 7 summarizes the partial and overall rank of the estimators.From Tables 1-7, we observe that: • The estimators  1 and  1 carry the property of consistency and tend to stable.
• For the NFWE distribution, the MPS and ML estimators have superior performances.

An application to the flood data
This section explores and demonstrates the applicability and superior performance of the NFWE distribution using an extreme value data.To establish the exceptional performance of the NFWE distribution over the other rival distributions, we examine a data set representing the maximum levels of the flood.

Description of the flood data set
Here, we provide the basic description of the flood data t that is considered to prove and establish the applicability of the NFWE distribution.The considered data set represents the maximum levels of the flood and is available at: https://data .world/datasets / flood.For interested readers, the data set is also provided in  .
Table 8 presents some useful key quantities of the flood data.From the key quantities, we can see the skewness is greater than 0, showing that the data is righted-skewed with a long right tail.Whereas, the value of kurtosis is 5.338 which ensures that the distribution of the flood data is Platykurtic.In Table 8, the terms  1 ,  2 , and  3 represent the 1  quartile, 2  quartile, and 3  quartile, respectively.
We show the graphical illustration of the flood data set in Fig. 3(a-d).In this regard, the histogram and box plot of the flood data are sketched.The TTT plot of the flood data is also sketched.The plots, provided in Fig. 3(a-d), show the heavy-tailed characteristics of the flood data that experienced extreme observations.

Competing distributions
For the comparative purpose, the proposed NFWE distribution is applied in comparison with the two-parameter rival models and three-parameter rival models.The two-parameter distributions include the FWE and Weibull distributions.Whereas, the three-parameter competitive distributions include the Marshall-Olkin Weibull (MO-Weibull), exponentiated Weibull (E-Weibull), and Marshall-Olkin Weibull (MO-Weibull), and alpha power transformed Weibull (APT-Weibull) distributions.The PDFs of these two-parameter and three-parameter competitive models are given by • Weibull distribution   • MO-Weibull distribution

Decisive tools
In this subsection, we discuss some decisive tools (i.e., selection criteria) to prove which probability distribution offers the best fit to the flood data set.The decisive tools are • CAIC • BIC  log () − 2.
• The AD test . • The KS test statistic Besides the above decisive tools, the p-value associated with the KS statistic is also calculated.It is a very important statistical measure that indicates the efficiency of the competing models.The higher p-value represents the higher efficiency of the model, in other words, the model with the highest p-value, provides the best fit (or adequate or reasonable fit) to the data.

Analysis of the flood data
We apply the NFWE distribution and its competing distributions to analyze flood data.The results of these distributions for the flood data set are prevailed by utilizing the - with () and  = ""; see  .
After performing the analysis using the flood data set, Table 9 presents the estimated parameters values.Whereas, the values of the decisive tools are presented in Tables 10 and 11.For the NFWE distribution, the p-value shows the highest value and the fitting criteria shows the lowest numerical values.These facts provide concrete evidence that the NFWE distribution is the best model for the flood data.
The best fitting claim of the NFWE distribution in Tables 10 and 11, is also supported visually in Fig. 4(a-f) and Fig. 5(a-f).The plots in Fig. 4(a-f) and Fig. 5(a-f) also confirm the suitability of the NFWE distribution for the flood data.

Final remarks
This paper considered a NFWE distribution as a potential modification of the FWE model using the weighted T-X procedure.For the NFWE model, some distributional characteristics were derived.The estimators of the FWE distribution were obtained through different estimation methods.For the NFWE distribution, the best estimation method was identified using a comprehensive simulation study.The practical evaluation of the NFWE distribution was demonstrated by considering the flood data with extreme observations.By deciding on different comparative tests, there were concrete evidence that the NFWE distribution is the most appropriate model (i.e., an adequate competing model) for the flood and other extreme event data.
The future work includes the development of (i) bivariate and multivariate modifications of the NFWE model and their practical implementations, (ii) regression structure of the NFWE model for survival data, and (iii) neutrosophic extension of the NFWE model for environmental studies.

Declaration of competing interest
The authors declare no conflict of interest.

Table 7
The partial and overall ranks of the estimators.

Table 8
The BMs of the flood data.